SEARCH · Search NASA

Results for “multivariate dual numbers”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

4 records

All sources NASA NTRS DOE OSTI

DOE OSTI2023

Efficient GPU Implementation of Automatic Differentiation for Computational Fluid Dynamics

Many scientific and engineering applications require repeated calculations of derivatives of output functions with respect to input parameters. Automatic Differentiation (AD) is a method that automates derivative calculations and can significantly speed up code development. In Computational Fluid Dynamics (CFD), derivatives of flux functions with respect to state variables (Jacobian) are needed for efficient solutions of the nonlinear governing equations. AD of flux functions on graphics processing units (GPUs) is challenging as flux computations involve many intermediate variables that create high register pressure and require significant memory traffic because of the need to store the derivatives. This paper presents a forward-mode AD method based on multivariate dual numbers that addresses these challenges and simultaneously reduces the floating-point operation count. The dimension of the multivariate dual numbers is optimized for performance. The flux computations are restructured to minimize the number of temporary variables and reduce register pressure. For effective utilization of memory bandwidth, shared memory is used to store the local flux Jacobian. This AD implementation is compared with several other Jacobian implementations on an NVIDIA V100 GPU (V100). For three-dimensional perfect-gas compressible-flow equations implemented in a practical CFD code, the AD implementation of a flux Jacobian based on multivariate dual numbers of dimension 5 outperforms all other GPU AD implementations on V100. Its performance is comparable with the optimized hand-differentiated version. Finally, the implementation achieves 75% of the peak floating-point throughput and 61 % of the peak global device memory bandwidth usage.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2024

Calculation of machine precision second order derivatives using dual-complex numbers

It is well known that both complex and dual numbers can be employed to obtain machine precision first-order derivatives; however, neither, on their own, can compute machine precision 2nd order derivatives. To address this limitation, it is demonstrated in this paper that combined dual-complex numbers can be used to compute machine precision 1st and 2nd order derivatives. The dual-complex approach is simpler than utilizing multicomplex or hyper-dual numbers as existing dual libraries can be used as is or easily augmented to accept complex numbers, and the complexity of developing, integrating, and deploying multicomplex or hyper-dual libraries is avoided. The efficacy of this approach is demonstrated for both univariate and multivariate functions. Finally, source code examples using the Python, Julia, and Mathematica languages are provided as supplemental material.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2006

Multiple-Flat-Panel System Displays Multidimensional Data

The NASA Ames hyperwall is a display system designed to facilitate the visualization of sets of multivariate and multidimensional data like those generated in complex engineering and scientific computations. The hyperwall includes a 77 matrix of computer-driven flat-panel video display units, each presenting an image of 1,280 1,024 pixels. The term hyperwall reflects the fact that this system is a more capable successor to prior computer-driven multiple-flat-panel display systems known by names that include the generic term powerwall and the trade names PowerWall and Powerwall. Each of the 49 flat-panel displays is driven by a rack-mounted, dual-central-processing- unit, workstation-class personal computer equipped with a hig-hperformance graphical-display circuit card and with a hard-disk drive having a storage capacity of 100 GB. Each such computer is a slave node in a master/ slave computing/data-communication system (see Figure 1). The computer that acts as the master node is similar to the slave-node computers, except that it runs the master portion of the system software and is equipped with a keyboard and mouse for control by a human operator. The system utilizes commercially available master/slave software along with custom software that enables the human controller to interact simultaneously with any number of selected slave nodes. In a powerwall, a single rendering task is spread across multiple processors and then the multiple outputs are tiled into one seamless super-display. It must be noted that the hyperwall concept subsumes the powerwall concept in that a single scene could be rendered as a mosaic image on the hyperwall. However, the hyperwall offers a wider set of capabilities to serve a different purpose: The hyperwall concept is one of (1) simultaneously displaying multiple different but related images, and (2) providing means for composing and controlling such sets of images. In place of elaborate software or hardware crossbar switches, the hyperwall concept substitutes reliance on the human visual system for integration, synthesis, and discrimination of patterns in complex and high-dimensional data spaces represented by the multiple displayed images. The variety of multidimensional data sets that can be displayed on the hyperwall is practically unlimited. For example, Figure 2 shows a hyperwall display of surface pressures and streamlines from a computational simulation of airflow about an aerospacecraft at various Mach numbers and angles of attack. In this display, Mach numbers increase from left to right and angles of attack increase from bottom to top. That is, all images in the same column represent simulations at the same Mach number, while all images in the same row represent simulations at the same angle of attack. The same viewing transformations and the same mapping from surface pressure to colors were used in generating all the images.

Gundo, Daniel↗

NASA NTRS2013

A Framework for Robust Multivariable Optimization of Integrated Circuits in Space Applications

Application Specific Integrated Circuit (ASIC) design for space applications involves multiple challenges of maximizing performance, minimizing power and ensuring reliable operation in extreme environments. This is a complex multidimensional optimization problem which must be solved early in the development cycle of a system due to the time required for testing and qualification severely limiting opportunities to modify and iterate. Manual design techniques which generally involve simulation at one or a small number of corners with a very limited set of simultaneously variable parameters in order to make the problem tractable are inefficient and not guaranteed to achieve the best possible results within the performance envelope defined by the process and environmental requirements. What is required is a means to automate design parameter variation, allow the designer to specify operational constraints and performance goals, and to analyze the results in a way which facilitates identifying the tradeoffs defining the performance envelope over the full set of process and environmental corner cases. The system developed by the Mixed Signal ASIC Group (MSAG) at the Goddard Space Flight Center is implemented as framework of software modules, templates and function libraries. It integrates CAD tools and a mathematical computing environment, and can be customized for new circuit designs with only a modest amount of effort as most common tasks are already encapsulated. Customization is required for simulation test benches to determine performance metrics and for cost function computation. Templates provide a starting point for both while toolbox functions minimize the code required. Once a test bench has been coded to optimize a particular circuit, it is also used to verify the final design. The combination of test bench and cost function can then serve as a template for similar circuits or be re-used to migrate the design to different processes by re-running it with the new process specific device models. The system has been used in the design of time to digital converters for laser ranging and time-of-flight mass spectrometry to optimize analog, mixed signal and digital circuits such as charge sensitive amplifiers, comparators, delay elements, radiation tolerant dual interlocked (DICE) flip-flops and two of three voter gates.

DuMonthier, Jeffrey↗